Finding the General solutions. I think using the ricatti method Problem 3.1: Find the GS ofxyy' – y? = x/x? + (ay)²where a + 0 is any number.

given differential equation xyy'-y2=xx2+αy2now divide both sides by xy ,we gety'-yx=x2y2+α…

Answered: Problem 3.1: Find the GS of xyy' – y? =…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter5: Inverse, Exponential, And Logarithmic Functions

Section5.4: Logarithmic Functions

Problem 45E

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Finding the General solutions. I think using the ricatti method

Problem 3.1: Find the GS of
xyy' – y? = x/x? + (ay)²
where a + 0 is any number.

Expert Solution

Step 1

$g i v e n d i f f e r e n t i a l e q u a t i o n x y y' - y^{2} = x \sqrt{x^{2} + {(α y)}^{2}} n o w d i v i d e b o t h s i d e s b y x y, w e g e t y' - \frac{y}{x} = \sqrt{\frac{x^{2}}{y^{2}} + α} . . . . . (a) p u t y = v x . . . . (i) \Rightarrow v = \frac{y}{x} n o w d i f f e r e n t i a t e (i) w . r . t o x, w e g e t y' = v + x v' p u t t h e v a l u e o f y' i n e q u a t i o n (a) v + x v' - v = \sqrt{\frac{1}{v^{2}} + α} x v' = \sqrt{\frac{1 + α v^{2}}{v^{2}}} \frac{v}{\sqrt{1 + α v^{2}}} d v = \frac{d x}{x} i n t e g r a t e b o t h s i d e s w e g e t \frac{\sqrt{1 + α v^{2}}}{α} = \log x + k \frac{\sqrt{1 + α \frac{y^{2}}{x^{2}}}}{α} = \log x + k$

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